Arbitrarily weak head-on collision can induce annihilation -- The role of hidden instabilities

Abstract

In this paper, we focus on annihilation dynamics for the head-on collision of traveling patterns. A representative and well-known example of annihilation is the one observed for 1-dimensional traveling pulses of the FitzHugh-Nagumo equations. In this paper, we present a new and completely different type of annihilation arising in a class of three-component reaction diffusion system. It is even counterintuitive in the sense that the two traveling spots or pulses come together very slowly but do not merge, keeping some separation, and then they start to repel each other for a certain time. Finally, up and down oscillatory instability emerges and grows enough for patterns to become extinct eventually (see Figs. 1-3). There is a kind of hidden instability embedded in the traveling patterns, which causes the above annihilation dynamics. The hidden instability here turns out to be a codimension 2 singularity consisting of drift and Hopf (DH) instabilities, and there is a parameter regime emanating from the codimension 2 point in which a new type of annihilation is observed. The above scenario can be proved analytically up to the onset of annihilation by reducing it to a finite-dimensional system. Transition from preservation to annihilation is also discussed in this framework.